Liquid-cooled internal combustion engines are commonly equipped with a sensor such as thermistor that is responsive to the temperature of the liquid coolant. In a typical motor vehicle application, the measured engine coolant temperature (ECT) is used for numerous control and diagnostic purposes such as identifying a cold-start condition of the engine. Since these control and diagnostic purposes can be frustrated by malfunctions of the ECT sensor, various diagnostic routines have been developed for verifying proper operation of the ECT sensor. One type of sensor malfunction that is particularly difficult to reliably diagnose is a low range limiting condition where the sensor output remains fixed for coolant temperatures below a certain value. This situation is graphically illustrated in FIG. 1, where the solid trace and dashed traces respectively represent the actual and measured coolant temperature over a period of time during which the engine is turned off and subsequently restarted. For example, the engine is turned off at time t0 and then restarted at time t2. In the illustration, the temperature sensor providing the measured coolant temperature is low range limited to approximately 70° C. In other words, the measured temperature faithfully tracks the actual coolant temperature until time t1 when the actual coolant temperature falls below 70° C. The measured coolant temperature remains fixed at 70° C. until time t3 when the actual coolant temperature rises above 70° C., whereafter the measured temperature faithfully tracks the actual coolant temperature. When this sort of temperature sensor malfunction occurs, the engine controller can fail to recognize a cold-start condition of the engine, for example, and various inappropriate control responses may result. Accordingly, what is needed is a method of diagnosing low range limiting malfunctions of an ECT sensor so that the driver of the vehicle may be prompted to have the vehicle serviced when such malfunctions occur.